Cuscuton and low-energy limit of Hořava-Lifshitz gravity

Hořava-Lifshitz theory has been recently put forth as a proposal for a renormalizable theory of quantum gravity [1]. It explicitly breaks Lorentz invariance, introducing an apparent extra scalar degree of freedom. I show that the low energy limit of (non-projectible) Hořava-Lifshitz gravity is uniquely given by the quadratic cuscuton model: a covariant scalar field theory with an infinite speed of sound and a quadratic potential, minimally coupled to Einstein gravity. This implies that the extra scalar is nondynamical to all orders in perturbation theory. Cosmological constraints on the quadratic cuscuton model constrain the low energy Lorentz breaking parameter of Hořava-Lifshitz theory to $|\lambda -1|<0.014$. We also point out that the spatial hypersurfaces are constant mean curvature or uniform expansion surfaces and introduce geometrical symmetries that can protect the nondynamical nature of these theories.

Figure 1

A cartoon of constant field hypersurfaces of cuscuton in a fixed space-time. As we argue in the text, different surfaces are decoupled in the action, and are thus fixed by boundary conditions at spatial infinity.